Existence and Nonexistence of Solution for a Class of Quasilinear Schrödinger Equations with Critical Growth

In this work, we study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations: − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) + h ( x ) g ( u ) in R N , where N ≥ 3 , g : R → R + is a continuously differentiable function, V ( x ) is a potential that can change sign, the function h ( x ) belongs to L 2 N / ( N + 2 ) ( R N ) and the nonlinearity f ( x , s ) is possibly discontinuous and may exhibit critical growth. In order to obtain the nonexistence result, we deduce a Pohozaev identity and the existence of solution is proved by means of a fixed point theorem.